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Probability Theory and Related Fields

, Volume 125, Issue 2, pp 195–224 | Cite as

Sharp metastability threshold for two-dimensional bootstrap percolation

  • Alexander E. Holroyd

Abstract.

 In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p→0 and L→∞ simultaneously of the probability I(L,p) that the entire square is eventually active. We prove that I(L,p)→1 if \(\), and I(L,p)→0 if \(\), where λ=π2/18. We prove the same behaviour, with the same threshold λ, for the probability J(L,p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ2/6. The existence of the thresholds λ,λ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2].

Keywords

Numerical Prediction Percolation Model Infinite Lattice Bootstrap Percolation Inactive Site 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alexander E. Holroyd
    • 1
  1. 1.Department of Mathematics, UC Berkeley, CA 94720-3840. e-mail: holroyd@math.berkeley.eduUS

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